A SHARP FREIMAN TYPE ESTIMATE FOR SEMISUMS IN TWO AND THREE DIMENSIONAL EUCLIDEAN SPACES

Freiman's theorem is a classical result in additive combinatorics concerning the approximate structure of sets of integers that contain a high proportion of their internal sums. As a consequence, one can deduce an estimate for sets of real numbers: If A subset of R and vertical bar 1/2(A + A)vertical bar - vertical bar A vertical bar << vertical bar A vertical bar, then A is close to its convex hull. In this paper we prove a sharp form of the analogous result in dimensions 2 and 3.

Automated summary

This paper proves a sharp form of the analogous result in dimensions 2 and 3, which is related to the approximate structure of sets of integers in Freiman's theorem concerning real numbers estimation.